Weakly hyperbolic group actions

In this thesis, we investigate a class of dynamical systems, the weakly hyperbolic group actions, that arise naturally in the study of higher rank lattice actions. Motivated by classical results from the theory of hyperbolic dynamical systems, we provide results of two sorts. First, we establish the ergodicity of weakly hyperbolic group actions, generalizing Anosov's classical theorem for Anosov flows and diffeomorphisms. Secondly, for weakly hyperbolic group actions of lattice subgroups of higher rank noncompact simple Lie groups on tori, we show that weak hyperbolicity persists in the induced action on first homology. This result is a natural analogue to Manning's theorem concerning the classification of Anosov diffeomorphisms on tori.